Properties

Label 76.3.b.b
Level $76$
Weight $3$
Character orbit 76.b
Analytic conductor $2.071$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 76.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.07085000914\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \( x^{14} - 2 x^{13} + x^{12} + 14 x^{11} - 42 x^{10} + 28 x^{9} + 132 x^{8} - 440 x^{7} + 528 x^{6} + 448 x^{5} - 2688 x^{4} + 3584 x^{3} + 1024 x^{2} - 8192 x + 16384 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} - \beta_{7} q^{3} + \beta_{5} q^{4} + \beta_{12} q^{5} + ( - \beta_{12} + \beta_{6}) q^{6} + ( - \beta_{13} - \beta_{8} - \beta_{2}) q^{7} + ( - \beta_{12} - \beta_{11} - \beta_{7} - \beta_{2} - 3) q^{8} + ( - \beta_{13} - \beta_{12} + \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} + 2 \beta_{5} + \cdots - 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} - \beta_{7} q^{3} + \beta_{5} q^{4} + \beta_{12} q^{5} + ( - \beta_{12} + \beta_{6}) q^{6} + ( - \beta_{13} - \beta_{8} - \beta_{2}) q^{7} + ( - \beta_{12} - \beta_{11} - \beta_{7} - \beta_{2} - 3) q^{8} + ( - \beta_{13} - \beta_{12} + \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} + 2 \beta_{5} + \cdots - 6) q^{9}+ \cdots + ( - 3 \beta_{13} + 3 \beta_{12} + 3 \beta_{11} + 3 \beta_{9} - 6 \beta_{8} - 4 \beta_{7} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 2 q^{2} + 2 q^{4} - 40 q^{8} - 68 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 2 q^{2} + 2 q^{4} - 40 q^{8} - 68 q^{9} - 12 q^{10} + 4 q^{12} + 54 q^{13} + 30 q^{14} + 58 q^{16} + 34 q^{17} + 36 q^{18} + 32 q^{20} - 38 q^{21} + 36 q^{22} - 98 q^{24} - 86 q^{25} - 16 q^{26} + 18 q^{28} + 54 q^{29} - 204 q^{30} + 72 q^{32} + 20 q^{33} - 82 q^{34} + 96 q^{36} + 100 q^{37} - 148 q^{40} + 224 q^{41} + 224 q^{42} - 96 q^{44} - 168 q^{45} + 46 q^{46} + 296 q^{48} - 220 q^{49} - 58 q^{50} - 288 q^{52} + 14 q^{53} - 128 q^{54} + 12 q^{56} + 38 q^{57} - 72 q^{58} + 188 q^{60} + 28 q^{61} + 396 q^{62} - 118 q^{64} - 472 q^{65} - 32 q^{66} + 30 q^{68} + 122 q^{69} + 156 q^{70} + 80 q^{72} + 70 q^{73} - 224 q^{74} + 228 q^{77} + 274 q^{78} - 348 q^{80} + 334 q^{81} - 400 q^{82} - 216 q^{84} + 48 q^{85} - 124 q^{86} + 472 q^{88} + 416 q^{90} + 126 q^{92} - 176 q^{93} - 88 q^{94} - 106 q^{96} + 308 q^{97} + 68 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} + x^{12} + 14 x^{11} - 42 x^{10} + 28 x^{9} + 132 x^{8} - 440 x^{7} + 528 x^{6} + 448 x^{5} - 2688 x^{4} + 3584 x^{3} + 1024 x^{2} - 8192 x + 16384 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{12} - 2 \nu^{11} + \nu^{10} + 14 \nu^{9} - 42 \nu^{8} + 28 \nu^{7} + 132 \nu^{6} - 440 \nu^{5} + 528 \nu^{4} + 448 \nu^{3} - 640 \nu^{2} + 3584 \nu + 1024 ) / 1024 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{13} + 14 \nu^{12} - 9 \nu^{11} + 94 \nu^{10} - 286 \nu^{9} + 844 \nu^{8} - 980 \nu^{7} - 1880 \nu^{6} + 8592 \nu^{5} - 8832 \nu^{4} - 3968 \nu^{3} + 59392 \nu^{2} - 119808 \nu + 114688 ) / 20480 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{13} + 2 \nu^{12} - \nu^{11} - 14 \nu^{10} + 42 \nu^{9} - 28 \nu^{8} - 132 \nu^{7} + 440 \nu^{6} - 528 \nu^{5} - 448 \nu^{4} + 2688 \nu^{3} - 3584 \nu^{2} - 1024 \nu + 8192 ) / 4096 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{13} + 3 \nu^{11} - 16 \nu^{10} + 14 \nu^{9} + 56 \nu^{8} - 188 \nu^{7} + 176 \nu^{6} + 352 \nu^{5} - 1504 \nu^{4} + 1792 \nu^{3} + 1792 \nu^{2} + 4096 ) / 2048 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{13} + 3 \nu^{11} - 16 \nu^{10} + 14 \nu^{9} + 56 \nu^{8} - 188 \nu^{7} + 176 \nu^{6} + 352 \nu^{5} - 1504 \nu^{4} + 1792 \nu^{3} + 1792 \nu^{2} - 8192 \nu + 6144 ) / 2048 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 9 \nu^{13} + 6 \nu^{12} - \nu^{11} + 86 \nu^{10} - 254 \nu^{9} + 476 \nu^{8} - 20 \nu^{7} - 3320 \nu^{6} + 4688 \nu^{5} - 2048 \nu^{4} - 16512 \nu^{3} + 48128 \nu^{2} - 64512 \nu - 22528 ) / 10240 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 21 \nu^{13} + 6 \nu^{12} - 91 \nu^{11} + 246 \nu^{10} - 354 \nu^{9} - 244 \nu^{8} + 2100 \nu^{7} - 1560 \nu^{6} - 112 \nu^{5} + 3392 \nu^{4} - 10112 \nu^{3} + 4608 \nu^{2} - 13312 \nu + 172032 ) / 20480 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 9 \nu^{13} - 166 \nu^{12} - 159 \nu^{11} + 1354 \nu^{10} - 2146 \nu^{9} - 1116 \nu^{8} + 14740 \nu^{7} - 31240 \nu^{6} + 2992 \nu^{5} + 98048 \nu^{4} - 217728 \nu^{3} + \cdots - 1052672 ) / 20480 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 19 \nu^{13} - 126 \nu^{12} - 189 \nu^{11} + 1074 \nu^{10} - 1006 \nu^{9} - 2556 \nu^{8} + 15020 \nu^{7} - 22280 \nu^{6} - 11408 \nu^{5} + 107968 \nu^{4} - 167808 \nu^{3} + \cdots - 991232 ) / 20480 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 39 \nu^{13} + 114 \nu^{12} - 569 \nu^{11} + 674 \nu^{10} + 954 \nu^{9} - 7036 \nu^{8} + 12380 \nu^{7} + 1400 \nu^{6} - 49168 \nu^{5} + 99648 \nu^{4} - 57728 \nu^{3} - 163328 \nu^{2} + \cdots - 69632 ) / 20480 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 11 \nu^{13} - 38 \nu^{12} + 27 \nu^{11} + 106 \nu^{10} - 318 \nu^{9} + 180 \nu^{8} + 1036 \nu^{7} - 3176 \nu^{6} + 2288 \nu^{5} + 4288 \nu^{4} - 11904 \nu^{3} + 2560 \nu^{2} + 46080 \nu - 77824 ) / 4096 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 3 \nu^{13} + 9 \nu^{12} - \nu^{11} - 39 \nu^{10} + 112 \nu^{9} - 82 \nu^{8} - 328 \nu^{7} + 1012 \nu^{6} - 952 \nu^{5} - 1168 \nu^{4} + 4672 \nu^{3} - 4736 \nu^{2} - 7680 \nu + 12288 ) / 1024 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 43 \nu^{13} - 18 \nu^{12} + 293 \nu^{11} - 578 \nu^{10} - 98 \nu^{9} + 3292 \nu^{8} - 8140 \nu^{7} + 3400 \nu^{6} + 22416 \nu^{5} - 56256 \nu^{4} + 50816 \nu^{3} + 78336 \nu^{2} + \cdots + 57344 ) / 10240 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{5} + \beta_{4} + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - 2\beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{13} + \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{4} + 2 \beta_{3} + \beta_{2} + \beta _1 - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{13} - 2 \beta_{12} - \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} - 3 \beta_{7} - \beta_{6} - \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} + \beta _1 + 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3 \beta_{13} + 4 \beta_{12} + 3 \beta_{11} + 3 \beta_{10} + \beta_{9} + \beta_{8} + 3 \beta_{7} - 7 \beta_{6} - 5 \beta_{5} + 6 \beta_{4} - 4 \beta_{3} + 11 \beta_{2} - 7 \beta _1 + 18 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( \beta_{13} + 2 \beta_{12} - 5 \beta_{11} - 3 \beta_{10} - 3 \beta_{9} + 7 \beta_{8} + 9 \beta_{7} - 17 \beta_{6} - 5 \beta_{5} - 2 \beta_{4} - 6 \beta_{3} + 3 \beta_{2} + \beta _1 - 20 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 7 \beta_{13} + 4 \beta_{12} - \beta_{11} - \beta_{10} + 17 \beta_{9} - 11 \beta_{8} + 39 \beta_{7} + 9 \beta_{6} + 15 \beta_{5} + 2 \beta_{4} - 40 \beta_{3} - 25 \beta_{2} + 9 \beta _1 + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 23 \beta_{13} - 54 \beta_{12} - 53 \beta_{11} - 35 \beta_{10} + 29 \beta_{9} - 33 \beta_{8} - 23 \beta_{7} + 39 \beta_{6} - \beta_{5} - 34 \beta_{4} + 82 \beta_{3} - 53 \beta_{2} + 29 \beta _1 - 188 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 69 \beta_{13} + 52 \beta_{12} + 19 \beta_{11} - 61 \beta_{10} + 29 \beta_{9} - 15 \beta_{8} - 37 \beta_{7} + 5 \beta_{6} + 159 \beta_{5} - 34 \beta_{4} - 56 \beta_{3} + 59 \beta_{2} - 11 \beta _1 + 254 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 75 \beta_{13} + 210 \beta_{12} + 127 \beta_{11} - 103 \beta_{10} - 55 \beta_{9} + 115 \beta_{8} + 149 \beta_{7} - 101 \beta_{6} - 133 \beta_{5} + 38 \beta_{4} + 490 \beta_{3} + 255 \beta_{2} - 31 \beta _1 - 316 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 23 \beta_{13} + 340 \beta_{12} + 223 \beta_{11} - 97 \beta_{10} + 33 \beta_{9} + 5 \beta_{8} + 311 \beta_{7} + 57 \beta_{6} - 189 \beta_{5} - 130 \beta_{4} - 888 \beta_{3} - 89 \beta_{2} + 281 \beta _1 + 1886 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 137 \beta_{13} - 86 \beta_{12} - 85 \beta_{11} + 61 \beta_{10} + 317 \beta_{9} - 481 \beta_{8} + 201 \beta_{7} + 775 \beta_{6} - 1513 \beta_{5} + 414 \beta_{4} - 1118 \beta_{3} - 373 \beta_{2} + 213 \beta _1 - 2684 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 781 \beta_{13} - 620 \beta_{12} - 405 \beta_{11} - 453 \beta_{10} - 75 \beta_{9} - 519 \beta_{8} - 1645 \beta_{7} + 189 \beta_{6} + 1567 \beta_{5} - 1290 \beta_{4} - 4648 \beta_{3} - 109 \beta_{2} + 909 \beta _1 - 6618 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(39\)
\(\chi(n)\) \(1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
39.1
−1.92254 + 0.551226i
−1.92254 0.551226i
−1.89728 + 0.632718i
−1.89728 0.632718i
−0.0607713 + 1.99908i
−0.0607713 1.99908i
0.645572 + 1.89294i
0.645572 1.89294i
0.711746 + 1.86907i
0.711746 1.86907i
1.57398 + 1.23393i
1.57398 1.23393i
1.94929 + 0.447510i
1.94929 0.447510i
−1.92254 0.551226i 0.644704i 3.39230 + 2.11950i −2.32715 0.355377 1.23947i 8.62924i −5.35350 5.94475i 8.58436 4.47404 + 1.28279i
39.2 −1.92254 + 0.551226i 0.644704i 3.39230 2.11950i −2.32715 0.355377 + 1.23947i 8.62924i −5.35350 + 5.94475i 8.58436 4.47404 1.28279i
39.3 −1.89728 0.632718i 5.34370i 3.19934 + 2.40089i 5.79268 −3.38106 + 10.1385i 5.87536i −4.55095 6.57943i −19.5551 −10.9903 3.66514i
39.4 −1.89728 + 0.632718i 5.34370i 3.19934 2.40089i 5.79268 −3.38106 10.1385i 5.87536i −4.55095 + 6.57943i −19.5551 −10.9903 + 3.66514i
39.5 −0.0607713 1.99908i 5.37609i −3.99261 + 0.242973i −5.82257 10.7472 0.326712i 5.45132i 0.728358 + 7.96677i −19.9023 0.353845 + 11.6398i
39.6 −0.0607713 + 1.99908i 5.37609i −3.99261 0.242973i −5.82257 10.7472 + 0.326712i 5.45132i 0.728358 7.96677i −19.9023 0.353845 11.6398i
39.7 0.645572 1.89294i 0.820457i −3.16647 2.44406i −2.38184 −1.55308 0.529664i 12.3764i −6.67066 + 4.41614i 8.32685 −1.53765 + 4.50868i
39.8 0.645572 + 1.89294i 0.820457i −3.16647 + 2.44406i −2.38184 −1.55308 + 0.529664i 12.3764i −6.67066 4.41614i 8.32685 −1.53765 4.50868i
39.9 0.711746 1.86907i 4.44946i −2.98683 2.66060i 4.97973 −8.31634 3.16688i 12.2628i −7.09872 + 3.68892i −10.7977 3.54430 9.30745i
39.10 0.711746 + 1.86907i 4.44946i −2.98683 + 2.66060i 4.97973 −8.31634 + 3.16688i 12.2628i −7.09872 3.68892i −10.7977 3.54430 + 9.30745i
39.11 1.57398 1.23393i 2.90118i 0.954817 3.88437i 3.66290 3.57986 + 4.56639i 1.93414i −3.29019 7.29209i 0.583162 5.76533 4.51977i
39.12 1.57398 + 1.23393i 2.90118i 0.954817 + 3.88437i 3.66290 3.57986 4.56639i 1.93414i −3.29019 + 7.29209i 0.583162 5.76533 + 4.51977i
39.13 1.94929 0.447510i 3.19988i 3.59947 1.74465i −3.90374 −1.43198 6.23749i 2.64664i 6.23566 5.01164i −1.23921 −7.60953 + 1.74696i
39.14 1.94929 + 0.447510i 3.19988i 3.59947 + 1.74465i −3.90374 −1.43198 + 6.23749i 2.64664i 6.23566 + 5.01164i −1.23921 −7.60953 1.74696i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 39.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.3.b.b 14
3.b odd 2 1 684.3.g.b 14
4.b odd 2 1 inner 76.3.b.b 14
8.b even 2 1 1216.3.d.d 14
8.d odd 2 1 1216.3.d.d 14
12.b even 2 1 684.3.g.b 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.3.b.b 14 1.a even 1 1 trivial
76.3.b.b 14 4.b odd 2 1 inner
684.3.g.b 14 3.b odd 2 1
684.3.g.b 14 12.b even 2 1
1216.3.d.d 14 8.b even 2 1
1216.3.d.d 14 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{14} + 97T_{3}^{12} + 3595T_{3}^{10} + 63443T_{3}^{8} + 539872T_{3}^{6} + 1940896T_{3}^{4} + 1665792T_{3}^{2} + 393984 \) acting on \(S_{3}^{\mathrm{new}}(76, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} - 2 T^{13} + T^{12} + 14 T^{11} + \cdots + 16384 \) Copy content Toggle raw display
$3$ \( T^{14} + 97 T^{12} + 3595 T^{10} + \cdots + 393984 \) Copy content Toggle raw display
$5$ \( (T^{7} - 66 T^{5} - 28 T^{4} + 1337 T^{3} + \cdots - 13312)^{2} \) Copy content Toggle raw display
$7$ \( T^{14} + 453 T^{12} + \cdots + 46106276299 \) Copy content Toggle raw display
$11$ \( T^{14} + 880 T^{12} + \cdots + 94488337600 \) Copy content Toggle raw display
$13$ \( (T^{7} - 27 T^{6} - 293 T^{5} + \cdots - 1265920)^{2} \) Copy content Toggle raw display
$17$ \( (T^{7} - 17 T^{6} - 1107 T^{5} + \cdots - 69101055)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 19)^{7} \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 683970816311296 \) Copy content Toggle raw display
$29$ \( (T^{7} - 27 T^{6} - 2901 T^{5} + \cdots - 112127472)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + 4048 T^{12} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{7} - 50 T^{6} - 1508 T^{5} + \cdots + 914894720)^{2} \) Copy content Toggle raw display
$41$ \( (T^{7} - 112 T^{6} + \cdots + 18882293760)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + 13244 T^{12} + \cdots + 58\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{14} + 8204 T^{12} + \cdots + 64\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( (T^{7} - 7 T^{6} - 6645 T^{5} + \cdots + 18228603200)^{2} \) Copy content Toggle raw display
$59$ \( T^{14} + 30513 T^{12} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{7} - 14 T^{6} + \cdots + 522558358600)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + 27889 T^{12} + \cdots + 29\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{14} + 37812 T^{12} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{7} - 35 T^{6} - 5907 T^{5} + \cdots - 77971926925)^{2} \) Copy content Toggle raw display
$79$ \( T^{14} + 38740 T^{12} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{14} + 62788 T^{12} + \cdots + 34\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{7} - 23640 T^{5} + \cdots - 88310345728)^{2} \) Copy content Toggle raw display
$97$ \( (T^{7} - 154 T^{6} + \cdots + 4410892984320)^{2} \) Copy content Toggle raw display
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